Rational points on elliptic curves / Joseph H. Silverman, John Tate.

Includes bibliographical references (pages [259]-262) and index.

Ch. I. Geometry and Arithmetic -- 1. Rational Points on Conics -- 2. The Geometry of Cubic Curves -- 3. Weierstrass Normal Form -- 4. Explicit Formulas for the Group Law -- Ch. II. Points of Finite Order -- 1. Points of Order Two and Three -- 2. Real and Complex Points on Cubic Curves -- 3. The Discriminant -- 4. Points of Finite Order Have Integer Coordinates -- 5. The Nagell-Lutz Theorem and Further Developments -- Ch. III. The Group of Rational Points -- 1. Heights and Descent -- 2. The Height of P + P[subscript 0] -- 3. The Height of 2P -- 4. A Useful Homomorphism -- 5. Mordell's Theorem -- 6. Examples and Further Developments -- 7. Singular Cubic Curves -- Ch. IV. Cubic Curves over Finite Fields -- 1. Rational Points over Finite Fields -- 2. A Theorem of Gauss -- 3. Points of Finite Order Revisited -- 4. A Factorization Algorithm Using Elliptic Curves -- Ch. V. Integer Points on Cubic Curves -- 1. How Many Integer Points? -- 2. Taxicabs and Sums of Two Cubes -- 3. Thue's Theorem and Diophantine Approximation -- 4. Construction of an Auxiliary Polynomial -- 5. The Auxiliary Polynomial Is Small -- 6. The Auxiliary Polynomial Does Not Vanish -- 7. Proof of the Diophantine Approximation Theorem -- 8. Further Developments -- Ch. VI. Complex Multiplication -- 1. Abelian Extensions of Q -- 2. Algebraic Points on Cubic Curves -- 3. A Galois Representation -- 4. Complex Multiplication -- 5. Abelian Extensions of Q(i). Appendix A: Projective Geometry -- 1. Homogeneous Coordinates and the Projective Plane -- 2. Curves in the Projective Plane -- 3. Intersections of Projective Curves -- 4. Intersection Multiplicities and a Proof of Bezout's Theorem -- 5. Reduction Modulo pages.